LMFDB - Embedded newform 177.8.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,8,Mod(1,177)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.2921495107$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-13.7979$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.7979 q^{2} +27.0000 q^{3} +35.7870 q^{4} -405.222 q^{5} -345.544 q^{6} -375.715 q^{7} +1180.14 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-12.7979 q^{2} +27.0000 q^{3} +35.7870 q^{4} -405.222 q^{5} -345.544 q^{6} -375.715 q^{7} +1180.14 q^{8} +729.000 q^{9} +5186.00 q^{10} -1868.92 q^{11} +966.249 q^{12} +143.937 q^{13} +4808.37 q^{14} -10941.0 q^{15} -19684.0 q^{16} -7997.65 q^{17} -9329.69 q^{18} -54944.6 q^{19} -14501.7 q^{20} -10144.3 q^{21} +23918.3 q^{22} -57227.2 q^{23} +31863.7 q^{24} +86079.9 q^{25} -1842.09 q^{26} +19683.0 q^{27} -13445.7 q^{28} +44123.3 q^{29} +140022. q^{30} -127609. q^{31} +100857. q^{32} -50460.9 q^{33} +102353. q^{34} +152248. q^{35} +26088.7 q^{36} -343781. q^{37} +703178. q^{38} +3886.30 q^{39} -478217. q^{40} +57315.2 q^{41} +129826. q^{42} +131635. q^{43} -66883.0 q^{44} -295407. q^{45} +732390. q^{46} +13310.0 q^{47} -531469. q^{48} -682381. q^{49} -1.10164e6 q^{50} -215937. q^{51} +5151.07 q^{52} -642722. q^{53} -251902. q^{54} +757328. q^{55} -443395. q^{56} -1.48351e6 q^{57} -564687. q^{58} +205379. q^{59} -391545. q^{60} -1.24234e6 q^{61} +1.63313e6 q^{62} -273896. q^{63} +1.22879e6 q^{64} -58326.4 q^{65} +645794. q^{66} +333473. q^{67} -286212. q^{68} -1.54514e6 q^{69} -1.94846e6 q^{70} +4.40870e6 q^{71} +860319. q^{72} -790210. q^{73} +4.39968e6 q^{74} +2.32416e6 q^{75} -1.96630e6 q^{76} +702181. q^{77} -49736.5 q^{78} +1.80576e6 q^{79} +7.97640e6 q^{80} +531441. q^{81} -733516. q^{82} +3.39335e6 q^{83} -363034. q^{84} +3.24082e6 q^{85} -1.68465e6 q^{86} +1.19133e6 q^{87} -2.20558e6 q^{88} +9.39299e6 q^{89} +3.78060e6 q^{90} -54079.2 q^{91} -2.04799e6 q^{92} -3.44544e6 q^{93} -170341. q^{94} +2.22648e7 q^{95} +2.72315e6 q^{96} +773470. q^{97} +8.73307e6 q^{98} -1.36244e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9}+O(q^{10}) $ 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9} + 3609 q^{10} + 15070 q^{11} + 36666 q^{12} + 13662 q^{13} + 20861 q^{14} + 18306 q^{15} + 60482 q^{16} + 71919 q^{17} + 17496 q^{18} + 56231 q^{19} + 143053 q^{20} + 83187 q^{21} + 274198 q^{22} + 150029 q^{23} + 110889 q^{24} + 399672 q^{25} + 182846 q^{26} + 354294 q^{27} + 434150 q^{28} + 591285 q^{29} + 97443 q^{30} + 426733 q^{31} + 1205630 q^{32} + 406890 q^{33} + 403548 q^{34} + 912879 q^{35} + 989982 q^{36} + 7703 q^{37} - 417859 q^{38} + 368874 q^{39} + 618020 q^{40} + 770959 q^{41} + 563247 q^{42} + 793050 q^{43} + 2591274 q^{44} + 494262 q^{45} - 4068019 q^{46} + 1410373 q^{47} + 1633014 q^{48} + 1637427 q^{49} + 1021549 q^{50} + 1941813 q^{51} - 3749190 q^{52} + 1037934 q^{53} + 472392 q^{54} + 331974 q^{55} - 391748 q^{56} + 1518237 q^{57} + 653724 q^{58} + 3696822 q^{59} + 3862431 q^{60} - 1374623 q^{61} + 5251718 q^{62} + 2246049 q^{63} + 5077197 q^{64} + 3257170 q^{65} + 7403346 q^{66} - 2436904 q^{67} + 14119909 q^{68} + 4050783 q^{69} + 5185580 q^{70} + 14289172 q^{71} + 2994003 q^{72} + 5482515 q^{73} + 14934154 q^{74} + 10791144 q^{75} + 3822912 q^{76} + 23157109 q^{77} + 4936842 q^{78} + 19786414 q^{79} + 31978143 q^{80} + 9565938 q^{81} + 9749509 q^{82} + 30227337 q^{83} + 11722050 q^{84} + 9946981 q^{85} + 44295864 q^{86} + 15964695 q^{87} + 39970897 q^{88} + 31061677 q^{89} + 2630961 q^{90} + 26377785 q^{91} + 4719698 q^{92} + 11521791 q^{93} + 44488296 q^{94} + 15534599 q^{95} + 32552010 q^{96} + 12084118 q^{97} + 42274744 q^{98} + 10986030 q^{99}+O(q^{100}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9 + 3609 * q^10 + 15070 * q^11 + 36666 * q^12 + 13662 * q^13 + 20861 * q^14 + 18306 * q^15 + 60482 * q^16 + 71919 * q^17 + 17496 * q^18 + 56231 * q^19 + 143053 * q^20 + 83187 * q^21 + 274198 * q^22 + 150029 * q^23 + 110889 * q^24 + 399672 * q^25 + 182846 * q^26 + 354294 * q^27 + 434150 * q^28 + 591285 * q^29 + 97443 * q^30 + 426733 * q^31 + 1205630 * q^32 + 406890 * q^33 + 403548 * q^34 + 912879 * q^35 + 989982 * q^36 + 7703 * q^37 - 417859 * q^38 + 368874 * q^39 + 618020 * q^40 + 770959 * q^41 + 563247 * q^42 + 793050 * q^43 + 2591274 * q^44 + 494262 * q^45 - 4068019 * q^46 + 1410373 * q^47 + 1633014 * q^48 + 1637427 * q^49 + 1021549 * q^50 + 1941813 * q^51 - 3749190 * q^52 + 1037934 * q^53 + 472392 * q^54 + 331974 * q^55 - 391748 * q^56 + 1518237 * q^57 + 653724 * q^58 + 3696822 * q^59 + 3862431 * q^60 - 1374623 * q^61 + 5251718 * q^62 + 2246049 * q^63 + 5077197 * q^64 + 3257170 * q^65 + 7403346 * q^66 - 2436904 * q^67 + 14119909 * q^68 + 4050783 * q^69 + 5185580 * q^70 + 14289172 * q^71 + 2994003 * q^72 + 5482515 * q^73 + 14934154 * q^74 + 10791144 * q^75 + 3822912 * q^76 + 23157109 * q^77 + 4936842 * q^78 + 19786414 * q^79 + 31978143 * q^80 + 9565938 * q^81 + 9749509 * q^82 + 30227337 * q^83 + 11722050 * q^84 + 9946981 * q^85 + 44295864 * q^86 + 15964695 * q^87 + 39970897 * q^88 + 31061677 * q^89 + 2630961 * q^90 + 26377785 * q^91 + 4719698 * q^92 + 11521791 * q^93 + 44488296 * q^94 + 15534599 * q^95 + 32552010 * q^96 + 12084118 * q^97 + 42274744 * q^98 + 10986030 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−12.7979	−1.13119	−0.565594	−	0.824684i	$-0.691353\pi$
$2$	−12.7979	−1.13119	−0.565594	+	0.824684i	$0.691353\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	35.7870	0.279586
$5$	−405.222	−1.44977	−0.724883	−	0.688872i	$-0.758106\pi$
$5$	−405.222	−1.44977	−0.724883	+	0.688872i	$0.758106\pi$
$6$	−345.544	−0.653092
$7$	−375.715	−0.414014	−0.207007	−	0.978339i	$-0.566372\pi$
$7$	−375.715	−0.414014	−0.207007	+	0.978339i	$0.566372\pi$
$8$	1180.14	0.814924
$9$	729.000	0.333333
$10$	5186.00	1.63996
$11$	−1868.92	−0.423366	−0.211683	−	0.977338i	$-0.567895\pi$
$11$	−1868.92	−0.423366	−0.211683	+	0.977338i	$0.567895\pi$
$12$	966.249	0.161419
$13$	143.937	0.0181706	0.00908532	−	0.999959i	$-0.497108\pi$
$13$	143.937	0.0181706	0.00908532	+	0.999959i	$0.497108\pi$
$14$	4808.37	0.468328
$15$	−10941.0	−0.837023
$16$	−19684.0	−1.20142
$17$	−7997.65	−0.394813	−0.197406	−	0.980322i	$-0.563252\pi$
$17$	−7997.65	−0.394813	−0.197406	+	0.980322i	$0.563252\pi$
$18$	−9329.69	−0.377063
$19$	−54944.6	−1.83776	−0.918878	−	0.394543i	$-0.870903\pi$
$19$	−54944.6	−1.83776	−0.918878	+	0.394543i	$0.870903\pi$
$20$	−14501.7	−0.405334
$21$	−10144.3	−0.239031
$22$	23918.3	0.478907
$23$	−57227.2	−0.980743	−0.490372	−	0.871513i	$-0.663139\pi$
$23$	−57227.2	−0.980743	−0.490372	+	0.871513i	$0.663139\pi$
$24$	31863.7	0.470496
$25$	86079.9	1.10182
$26$	−1842.09	−0.0205544
$27$	19683.0	0.192450
$28$	−13445.7	−0.115753
$29$	44123.3	0.335951	0.167975	−	0.985791i	$-0.446277\pi$
$29$	44123.3	0.335951	0.167975	+	0.985791i	$0.446277\pi$
$30$	140022.	0.946830
$31$	−127609.	−0.769333	−0.384667	−	0.923056i	$-0.625684\pi$
$31$	−127609.	−0.769333	−0.384667	+	0.923056i	$0.625684\pi$
$32$	100857.	0.544105
$33$	−50460.9	−0.244431
$34$	102353.	0.446607
$35$	152248.	0.600224
$36$	26088.7	0.0931953
$37$	−343781.	−1.11577	−0.557886	−	0.829918i	$-0.688387\pi$
$37$	−343781.	−1.11577	−0.557886	+	0.829918i	$0.688387\pi$
$38$	703178.	2.07885
$39$	3886.30	0.0104908
$40$	−478217.	−1.18145
$41$	57315.2	0.129875	0.0649376	−	0.997889i	$-0.479315\pi$
$41$	57315.2	0.129875	0.0649376	+	0.997889i	$0.479315\pi$
$42$	129826.	0.270389
$43$	131635.	0.252483	0.126241	−	0.992000i	$-0.459709\pi$
$43$	131635.	0.252483	0.126241	+	0.992000i	$0.459709\pi$
$44$	−66883.0	−0.118367
$45$	−295407.	−0.483255
$46$	732390.	1.10941
$47$	13310.0	0.0186998	0.00934989	−	0.999956i	$-0.497024\pi$
$47$	13310.0	0.0186998	0.00934989	+	0.999956i	$0.497024\pi$
$48$	−531469.	−0.693639
$49$	−682381.	−0.828592
$50$	−1.10164e6	−1.24637
$51$	−215937.	−0.227945
$52$	5151.07	0.00508026
$53$	−642722.	−0.593004	−0.296502	−	0.955032i	$-0.595820\pi$
$53$	−642722.	−0.593004	−0.296502	+	0.955032i	$0.595820\pi$
$54$	−251902.	−0.217697
$55$	757328.	0.613782
$56$	−443395.	−0.337390
$57$	−1.48351e6	−1.06103
$58$	−564687.	−0.380023
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−391545.	−0.234020
$61$	−1.24234e6	−0.700785	−0.350393	−	0.936603i	$-0.613952\pi$
$61$	−1.24234e6	−0.700785	−0.350393	+	0.936603i	$0.613952\pi$
$62$	1.63313e6	0.870261
$63$	−273896.	−0.138005
$64$	1.22879e6	0.585932
$65$	−58326.4	−0.0263432
$66$	645794.	0.276497
$67$	333473.	0.135456	0.0677282	−	0.997704i	$-0.478425\pi$
$67$	333473.	0.135456	0.0677282	+	0.997704i	$0.478425\pi$
$68$	−286212.	−0.110384
$69$	−1.54514e6	−0.566233
$70$	−1.94846e6	−0.678966
$71$	4.40870e6	1.46186	0.730930	−	0.682452i	$-0.239086\pi$
$71$	4.40870e6	1.46186	0.730930	+	0.682452i	$0.239086\pi$
$72$	860319.	0.271641
$73$	−790210.	−0.237746	−0.118873	−	0.992909i	$-0.537928\pi$
$73$	−790210.	−0.237746	−0.118873	+	0.992909i	$0.537928\pi$
$74$	4.39968e6	1.26215
$75$	2.32416e6	0.636137
$76$	−1.96630e6	−0.513810
$77$	702181.	0.175280
$78$	−49736.5	−0.0118671
$79$	1.80576e6	0.412065	0.206032	−	0.978545i	$-0.433945\pi$
$79$	1.80576e6	0.412065	0.206032	+	0.978545i	$0.433945\pi$
$80$	7.97640e6	1.74177
$81$	531441.	0.111111
$82$	−733516.	−0.146913
$83$	3.39335e6	0.651411	0.325705	−	0.945471i	$-0.394398\pi$
$83$	3.39335e6	0.651411	0.325705	+	0.945471i	$0.394398\pi$
$84$	−363034.	−0.0668297
$85$	3.24082e6	0.572386
$86$	−1.68465e6	−0.285605
$87$	1.19133e6	0.193961
$88$	−2.20558e6	−0.345011
$89$	9.39299e6	1.41234	0.706169	−	0.708043i	$-0.250422\pi$
$89$	9.39299e6	1.41234	0.706169	+	0.708043i	$0.250422\pi$
$90$	3.78060e6	0.546653
$91$	−54079.2	−0.00752291
$92$	−2.04799e6	−0.274202
$93$	−3.44544e6	−0.444175
$94$	−170341.	−0.0211530
$95$	2.22648e7	2.66432
$96$	2.72315e6	0.314139
$97$	773470.	0.0860483	0.0430242	−	0.999074i	$-0.486301\pi$
$97$	773470.	0.0860483	0.0430242	+	0.999074i	$0.486301\pi$
$98$	8.73307e6	0.937293
$99$	−1.36244e6	−0.141122
$100$	3.08054e6	0.308054
$101$	−654307.	−0.0631912	−0.0315956	−	0.999501i	$-0.510059\pi$
$101$	−654307.	−0.0631912	−0.0315956	+	0.999501i	$0.510059\pi$
$102$	2.76354e6	0.257849
$103$	5.35811e6	0.483149	0.241574	−	0.970382i	$-0.422336\pi$
$103$	5.35811e6	0.483149	0.241574	+	0.970382i	$0.422336\pi$
$104$	169865.	0.0148077
$105$	4.11069e6	0.346539
$106$	8.22551e6	0.670799
$107$	−4.95592e6	−0.391094	−0.195547	−	0.980694i	$-0.562648\pi$
$107$	−4.95592e6	−0.391094	−0.195547	+	0.980694i	$0.562648\pi$
$108$	704395.	0.0538063
$109$	1.10724e7	0.818931	0.409465	−	0.912326i	$-0.365715\pi$
$109$	1.10724e7	0.818931	0.409465	+	0.912326i	$0.365715\pi$
$110$	−9.69223e6	−0.694303
$111$	−9.28208e6	−0.644191
$112$	7.39558e6	0.497404
$113$	1.00502e7	0.655239	0.327619	−	0.944810i	$-0.393754\pi$
$113$	1.00502e7	0.655239	0.327619	+	0.944810i	$0.393754\pi$
$114$	1.89858e7	1.20022
$115$	2.31897e7	1.42185
$116$	1.57904e6	0.0939270
$117$	104930.	0.00605688
$118$	−2.62843e6	−0.147268
$119$	3.00484e6	0.163458
$120$	−1.29119e7	−0.682110
$121$	−1.59943e7	−0.820761
$122$	1.58993e7	0.792720
$123$	1.54751e6	0.0749835
$124$	−4.56673e6	−0.215095
$125$	−3.22348e6	−0.147618
$126$	3.50530e6	0.156109
$127$	−2.50572e7	−1.08547	−0.542737	−	0.839903i	$-0.682612\pi$
$127$	−2.50572e7	−1.08547	−0.542737	+	0.839903i	$0.682612\pi$
$128$	−2.86357e7	−1.20690
$129$	3.55414e6	0.145771
$130$	746457.	0.0297991
$131$	−1.29122e7	−0.501823	−0.250912	−	0.968010i	$-0.580730\pi$
$131$	−1.29122e7	−0.501823	−0.250912	+	0.968010i	$0.580730\pi$
$132$	−1.80584e6	−0.0683394
$133$	2.06435e7	0.760857
$134$	−4.26777e6	−0.153227
$135$	−7.97598e6	−0.279008
$136$	−9.43831e6	−0.321742
$137$	3.21499e6	0.106821	0.0534106	−	0.998573i	$-0.482991\pi$
$137$	3.21499e6	0.106821	0.0534106	+	0.998573i	$0.482991\pi$
$138$	1.97745e7	0.640515
$139$	1.39325e7	0.440025	0.220012	−	0.975497i	$-0.429390\pi$
$139$	1.39325e7	0.440025	0.220012	+	0.975497i	$0.429390\pi$
$140$	5.44849e6	0.167814
$141$	359371.	0.0107963
$142$	−5.64222e7	−1.65364
$143$	−269007.	−0.00769284
$144$	−1.43497e7	−0.400473
$145$	−1.78797e7	−0.487050
$146$	1.01130e7	0.268935
$147$	−1.84243e7	−0.478388
$148$	−1.23029e7	−0.311954
$149$	4.45947e7	1.10441	0.552206	−	0.833708i	$-0.313786\pi$
$149$	4.45947e7	1.10441	0.552206	+	0.833708i	$0.313786\pi$
$150$	−2.97444e7	−0.719591
$151$	3.86098e7	0.912595	0.456298	−	0.889827i	$-0.349175\pi$
$151$	3.86098e7	0.912595	0.456298	+	0.889827i	$0.349175\pi$
$152$	−6.48421e7	−1.49763
$153$	−5.83029e6	−0.131604
$154$	−8.98647e6	−0.198274
$155$	5.17099e7	1.11535
$156$	139079.	0.00293309
$157$	6.75271e7	1.39261	0.696305	−	0.717746i	$-0.254826\pi$
$157$	6.75271e7	1.39261	0.696305	+	0.717746i	$0.254826\pi$
$158$	−2.31100e7	−0.466123
$159$	−1.73535e7	−0.342371
$160$	−4.08696e7	−0.788825
$161$	2.15011e7	0.406042
$162$	−6.80134e6	−0.125688
$163$	−3.73402e7	−0.675336	−0.337668	−	0.941265i	$-0.609638\pi$
$163$	−3.73402e7	−0.675336	−0.337668	+	0.941265i	$0.609638\pi$
$164$	2.05114e6	0.0363113
$165$	2.04478e7	0.354367
$166$	−4.34278e7	−0.736868
$167$	4.08725e7	0.679085	0.339542	−	0.940591i	$-0.389728\pi$
$167$	4.08725e7	0.679085	0.339542	+	0.940591i	$0.389728\pi$
$168$	−1.19717e7	−0.194792
$169$	−6.27278e7	−0.999670
$170$	−4.14758e7	−0.647476
$171$	−4.00546e7	−0.612585
$172$	4.71082e6	0.0705906
$173$	3.92808e7	0.576791	0.288395	−	0.957511i	$-0.406878\pi$
$173$	3.92808e7	0.576791	0.288395	+	0.957511i	$0.406878\pi$
$174$	−1.52466e7	−0.219406
$175$	−3.23415e7	−0.456170
$176$	3.67879e7	0.508640
$177$	5.54523e6	0.0751646
$178$	−1.20211e8	−1.59762
$179$	−4.89222e7	−0.637559	−0.318780	−	0.947829i	$-0.603273\pi$
$179$	−4.89222e7	−0.637559	−0.318780	+	0.947829i	$0.603273\pi$
$180$	−1.05717e7	−0.135111
$181$	−8.97873e7	−1.12549	−0.562743	−	0.826632i	$-0.690254\pi$
$181$	−8.97873e7	−1.12549	−0.562743	+	0.826632i	$0.690254\pi$
$182$	692102.	0.00850982
$183$	−3.35431e7	−0.404599
$184$	−6.75359e7	−0.799231
$185$	1.39307e8	1.61761
$186$	4.40945e7	0.502445
$187$	1.49470e7	0.167150
$188$	476326.	0.00522819
$189$	−7.39520e6	−0.0796771
$190$	−2.84943e8	−3.01384
$191$	1.17574e8	1.22094	0.610470	−	0.792040i	$-0.290981\pi$
$191$	1.17574e8	1.22094	0.610470	+	0.792040i	$0.290981\pi$
$192$	3.31773e7	0.338288
$193$	−1.89578e8	−1.89818	−0.949092	−	0.314999i	$-0.897996\pi$
$193$	−1.89578e8	−1.89818	−0.949092	+	0.314999i	$0.897996\pi$
$194$	−9.89881e6	−0.0973368
$195$	−1.57481e6	−0.0152092
$196$	−2.44204e7	−0.231663
$197$	6.19268e7	0.577094	0.288547	−	0.957466i	$-0.406828\pi$
$197$	6.19268e7	0.577094	0.288547	+	0.957466i	$0.406828\pi$
$198$	1.74365e7	0.159636
$199$	1.68475e8	1.51548	0.757741	−	0.652556i	$-0.226303\pi$
$199$	1.68475e8	1.51548	0.757741	+	0.652556i	$0.226303\pi$
$200$	1.01586e8	0.897901
$201$	9.00378e6	0.0782058
$202$	8.37378e6	0.0714812
$203$	−1.65778e7	−0.139088
$204$	−7.72772e6	−0.0637303
$205$	−2.32254e7	−0.188289
$206$	−6.85727e7	−0.546532
$207$	−4.17187e7	−0.326914
$208$	−2.83326e6	−0.0218305
$209$	1.02687e8	0.778044
$210$	−5.26084e7	−0.392001
$211$	1.90058e7	0.139283	0.0696413	−	0.997572i	$-0.477815\pi$
$211$	1.90058e7	0.139283	0.0696413	+	0.997572i	$0.477815\pi$
$212$	−2.30011e7	−0.165796
$213$	1.19035e8	0.844006
$214$	6.34255e7	0.442401
$215$	−5.33414e7	−0.366041
$216$	2.32286e7	0.156832
$217$	4.79445e7	0.318515
$218$	−1.41703e8	−0.926364
$219$	−2.13357e7	−0.137262
$220$	2.71025e7	0.171605
$221$	−1.15116e6	−0.00717400
$222$	1.18791e8	0.728701
$223$	−9.54195e7	−0.576196	−0.288098	−	0.957601i	$-0.593023\pi$
$223$	−9.54195e7	−0.576196	−0.288098	+	0.957601i	$0.593023\pi$
$224$	−3.78936e7	−0.225267
$225$	6.27522e7	0.367274
$226$	−1.28622e8	−0.741198
$227$	1.24600e7	0.0707012	0.0353506	−	0.999375i	$-0.488745\pi$
$227$	1.24600e7	0.0707012	0.0353506	+	0.999375i	$0.488745\pi$
$228$	−5.30902e7	−0.296649
$229$	−1.55581e8	−0.856118	−0.428059	−	0.903751i	$-0.640802\pi$
$229$	−1.55581e8	−0.856118	−0.428059	+	0.903751i	$0.640802\pi$
$230$	−2.96781e8	−1.60838
$231$	1.89589e7	0.101198
$232$	5.20715e7	0.273774
$233$	−1.59313e8	−0.825100	−0.412550	−	0.910935i	$-0.635362\pi$
$233$	−1.59313e8	−0.825100	−0.412550	+	0.910935i	$0.635362\pi$
$234$	−1.34289e6	−0.00685147
$235$	−5.39351e6	−0.0271103
$236$	7.34990e6	0.0363990
$237$	4.87556e7	0.237906
$238$	−3.84557e7	−0.184902
$239$	2.85477e8	1.35263	0.676314	−	0.736614i	$-0.263576\pi$
$239$	2.85477e8	1.35263	0.676314	+	0.736614i	$0.263576\pi$
$240$	2.15363e8	1.00561
$241$	2.08212e8	0.958177	0.479088	−	0.877767i	$-0.340967\pi$
$241$	2.08212e8	0.958177	0.479088	+	0.877767i	$0.340967\pi$
$242$	2.04694e8	0.928435
$243$	1.43489e7	0.0641500
$244$	−4.44595e7	−0.195930
$245$	2.76516e8	1.20127
$246$	−1.98049e7	−0.0848204
$247$	−7.90856e6	−0.0333932
$248$	−1.50596e8	−0.626948
$249$	9.16203e7	0.376092
$250$	4.12539e7	0.166984
$251$	−3.40393e8	−1.35870	−0.679348	−	0.733816i	$-0.737737\pi$
$251$	−3.40393e8	−1.35870	−0.679348	+	0.733816i	$0.737737\pi$
$252$	−9.80192e6	−0.0385842
$253$	1.06953e8	0.415214
$254$	3.20680e8	1.22788
$255$	8.75022e7	0.330467
$256$	2.09193e8	0.779304
$257$	1.59818e8	0.587298	0.293649	−	0.955913i	$-0.405130\pi$
$257$	1.59818e8	0.587298	0.293649	+	0.955913i	$0.405130\pi$
$258$	−4.54857e7	−0.164894
$259$	1.29164e8	0.461945
$260$	−2.08733e6	−0.00736518
$261$	3.21659e7	0.111984
$262$	1.65249e8	0.567656
$263$	−7.73432e7	−0.262166	−0.131083	−	0.991371i	$-0.541845\pi$
$263$	−7.73432e7	−0.262166	−0.131083	+	0.991371i	$0.541845\pi$
$264$	−5.95507e7	−0.199192
$265$	2.60445e8	0.859717
$266$	−2.64194e8	−0.860672
$267$	2.53611e8	0.815414
$268$	1.19340e7	0.0378717
$269$	−4.47623e8	−1.40210	−0.701050	−	0.713112i	$-0.747285\pi$
$269$	−4.47623e8	−1.40210	−0.701050	+	0.713112i	$0.747285\pi$
$270$	1.02076e8	0.315610
$271$	−3.39469e8	−1.03611	−0.518056	−	0.855346i	$-0.673344\pi$
$271$	−3.39469e8	−1.03611	−0.518056	+	0.855346i	$0.673344\pi$
$272$	1.57426e8	0.474335
$273$	−1.46014e6	−0.00434335
$274$	−4.11452e7	−0.120835
$275$	−1.60876e8	−0.466475
$276$	−5.52958e7	−0.158311
$277$	−3.30374e8	−0.933957	−0.466979	−	0.884269i	$-0.654658\pi$
$277$	−3.30374e8	−0.933957	−0.466979	+	0.884269i	$0.654658\pi$
$278$	−1.78307e8	−0.497751
$279$	−9.30268e7	−0.256444
$280$	1.79673e8	0.489137
$281$	−5.59715e7	−0.150486	−0.0752428	−	0.997165i	$-0.523973\pi$
$281$	−5.59715e7	−0.150486	−0.0752428	+	0.997165i	$0.523973\pi$
$282$	−4.59920e6	−0.0122127
$283$	−3.42839e8	−0.899160	−0.449580	−	0.893240i	$-0.648426\pi$
$283$	−3.42839e8	−0.899160	−0.449580	+	0.893240i	$0.648426\pi$
$284$	1.57774e8	0.408716
$285$	6.01149e8	1.53824
$286$	3.44273e6	0.00870205
$287$	−2.15342e7	−0.0537702
$288$	7.35251e7	0.181368
$289$	−3.46376e8	−0.844123
$290$	2.28824e8	0.550945
$291$	2.08837e7	0.0496800
$292$	−2.82792e7	−0.0664703
$293$	5.81815e8	1.35129	0.675644	−	0.737228i	$-0.263865\pi$
$293$	5.81815e8	1.35129	0.675644	+	0.737228i	$0.263865\pi$
$294$	2.35793e8	0.541147
$295$	−8.32241e7	−0.188743
$296$	−4.05708e8	−0.909269
$297$	−3.67860e7	−0.0814769
$298$	−5.70719e8	−1.24930
$299$	−8.23711e6	−0.0178207
$300$	8.31745e7	0.177855
$301$	−4.94572e7	−0.104531
$302$	−4.94125e8	−1.03232
$303$	−1.76663e7	−0.0364835
$304$	1.08153e9	2.20791
$305$	5.03422e8	1.01597
$306$	7.46156e7	0.148869
$307$	−6.21817e8	−1.22653	−0.613265	−	0.789877i	$-0.710144\pi$
$307$	−6.21817e8	−1.22653	−0.613265	+	0.789877i	$0.710144\pi$
$308$	2.51290e7	0.0490057
$309$	1.44669e8	0.278946
$310$	−6.61779e8	−1.26167
$311$	−7.86009e8	−1.48172	−0.740860	−	0.671659i	$-0.765582\pi$
$311$	−7.86009e8	−1.48172	−0.740860	+	0.671659i	$0.765582\pi$
$312$	4.58636e6	0.00854922
$313$	−3.72650e8	−0.686904	−0.343452	−	0.939170i	$-0.611596\pi$
$313$	−3.72650e8	−0.686904	−0.343452	+	0.939170i	$0.611596\pi$
$314$	−8.64207e8	−1.57530
$315$	1.10989e8	0.200075
$316$	6.46228e7	0.115208
$317$	2.38136e8	0.419873	0.209936	−	0.977715i	$-0.432674\pi$
$317$	2.38136e8	0.419873	0.209936	+	0.977715i	$0.432674\pi$
$318$	2.22089e8	0.387286
$319$	−8.24630e7	−0.142230
$320$	−4.97932e8	−0.849465
$321$	−1.33810e8	−0.225798
$322$	−2.75170e8	−0.459309
$323$	4.39428e8	0.725569
$324$	1.90187e7	0.0310651
$325$	1.23901e7	0.0200208
$326$	4.77877e8	0.763932
$327$	2.98953e8	0.472810
$328$	6.76397e7	0.105838
$329$	−5.00077e6	−0.00774197
$330$	−2.61690e8	−0.400856
$331$	6.49978e8	0.985147	0.492573	−	0.870271i	$-0.336056\pi$
$331$	6.49978e8	0.985147	0.492573	+	0.870271i	$0.336056\pi$
$332$	1.21438e8	0.182125
$333$	−2.50616e8	−0.371924
$334$	−5.23084e8	−0.768172
$335$	−1.35131e8	−0.196380
$336$	1.99681e8	0.287176
$337$	6.65064e8	0.946583	0.473292	−	0.880906i	$-0.343066\pi$
$337$	6.65064e8	0.946583	0.473292	+	0.880906i	$0.343066\pi$
$338$	8.02786e8	1.13081
$339$	2.71355e8	0.378302
$340$	1.15979e8	0.160031
$341$	2.38491e8	0.325710
$342$	5.12616e8	0.692949
$343$	5.65798e8	0.757063
$344$	1.55347e8	0.205754
$345$	6.26123e8	0.820905
$346$	−5.02712e8	−0.652459
$347$	2.66064e8	0.341848	0.170924	−	0.985284i	$-0.445325\pi$
$347$	2.66064e8	0.341848	0.170924	+	0.985284i	$0.445325\pi$
$348$	4.26341e7	0.0542288
$349$	1.20795e9	1.52111	0.760556	−	0.649273i	$-0.224927\pi$
$349$	1.20795e9	1.52111	0.760556	+	0.649273i	$0.224927\pi$
$350$	4.13904e8	0.516014
$351$	2.83311e6	0.00349694
$352$	−1.88494e8	−0.230356
$353$	2.38306e8	0.288352	0.144176	−	0.989552i	$-0.453947\pi$
$353$	2.38306e8	0.288352	0.144176	+	0.989552i	$0.453947\pi$
$354$	−7.09675e7	−0.0850253
$355$	−1.78650e9	−2.11936
$356$	3.36147e8	0.394870
$357$	8.11306e7	0.0943726
$358$	6.26103e8	0.721199
$359$	−1.03023e9	−1.17518	−0.587588	−	0.809160i	$-0.699922\pi$
$359$	−1.03023e9	−1.17518	−0.587588	+	0.809160i	$0.699922\pi$
$360$	−3.48620e8	−0.393816
$361$	2.12504e9	2.37734
$362$	1.14909e9	1.27314
$363$	−4.31846e8	−0.473866
$364$	−1.93533e6	−0.00210330
$365$	3.20210e8	0.344675
$366$	4.29282e8	0.457677
$367$	1.50357e9	1.58779	0.793893	−	0.608058i	$-0.208051\pi$
$367$	1.50357e9	1.58779	0.793893	+	0.608058i	$0.208051\pi$
$368$	1.12646e9	1.17828
$369$	4.17828e7	0.0432917
$370$	−1.78285e9	−1.82982
$371$	2.41480e8	0.245512
$372$	−1.23302e8	−0.124185
$373$	−5.53195e8	−0.551947	−0.275973	−	0.961165i	$-0.589000\pi$
$373$	−5.53195e8	−0.551947	−0.275973	+	0.961165i	$0.589000\pi$
$374$	−1.91290e8	−0.189079
$375$	−8.70340e7	−0.0852274
$376$	1.57076e7	0.0152389
$377$	6.35098e6	0.00610444
$378$	9.46432e7	0.0901297
$379$	1.05631e9	0.996680	0.498340	−	0.866982i	$-0.333943\pi$
$379$	1.05631e9	0.996680	0.498340	+	0.866982i	$0.333943\pi$
$380$	7.96789e8	0.744905
$381$	−6.76544e8	−0.626699
$382$	−1.50470e9	−1.38111
$383$	−1.04319e8	−0.0948782	−0.0474391	−	0.998874i	$-0.515106\pi$
$383$	−1.04319e8	−0.0948782	−0.0474391	+	0.998874i	$0.515106\pi$
$384$	−7.73164e8	−0.696807
$385$	−2.84539e8	−0.254115
$386$	2.42621e9	2.14720
$387$	9.59618e7	0.0841609
$388$	2.76802e7	0.0240579
$389$	−1.52386e9	−1.31257	−0.656284	−	0.754514i	$-0.727873\pi$
$389$	−1.52386e9	−1.31257	−0.656284	+	0.754514i	$0.727873\pi$
$390$	2.01543e7	0.0172045
$391$	4.57683e8	0.387210
$392$	−8.05302e8	−0.675239
$393$	−3.48629e8	−0.289728
$394$	−7.92534e8	−0.652802
$395$	−7.31734e8	−0.597398
$396$	−4.87577e7	−0.0394558
$397$	−2.77621e8	−0.222683	−0.111341	−	0.993782i	$-0.535515\pi$
$397$	−2.77621e8	−0.222683	−0.111341	+	0.993782i	$0.535515\pi$
$398$	−2.15614e9	−1.71429
$399$	5.57375e8	0.439281
$400$	−1.69440e9	−1.32375
$401$	3.39521e8	0.262943	0.131471	−	0.991320i	$-0.458030\pi$
$401$	3.39521e8	0.262943	0.131471	+	0.991320i	$0.458030\pi$
$402$	−1.15230e8	−0.0884654
$403$	−1.83676e7	−0.0139793
$404$	−2.34157e7	−0.0176674
$405$	−2.15352e8	−0.161085
$406$	2.12161e8	0.157335
$407$	6.42499e8	0.472380
$408$	−2.54834e8	−0.185758
$409$	2.23733e9	1.61696	0.808479	−	0.588525i	$-0.200291\pi$
$409$	2.23733e9	1.61696	0.808479	+	0.588525i	$0.200291\pi$
$410$	2.97237e8	0.212990
$411$	8.68047e7	0.0616733
$412$	1.91750e8	0.135082
$413$	−7.71639e7	−0.0539001
$414$	5.33912e8	0.369802
$415$	−1.37506e9	−0.944393
$416$	1.45171e7	0.00988674
$417$	3.76178e8	0.254049
$418$	−1.31418e9	−0.880114
$419$	1.72385e9	1.14485	0.572427	−	0.819955i	$-0.306002\pi$
$419$	1.72385e9	1.14485	0.572427	+	0.819955i	$0.306002\pi$
$420$	1.47109e8	0.0968875
$421$	2.37385e9	1.55048	0.775239	−	0.631668i	$-0.217629\pi$
$421$	2.37385e9	1.55048	0.775239	+	0.631668i	$0.217629\pi$
$422$	−2.43234e8	−0.157555
$423$	9.70301e6	0.00623326
$424$	−7.58499e8	−0.483253
$425$	−6.88436e8	−0.435013
$426$	−1.52340e9	−0.954729
$427$	4.66765e8	0.290135
$428$	−1.77357e8	−0.109344
$429$	−7.26318e6	−0.00444147
$430$	6.82659e8	0.414061
$431$	−2.92636e9	−1.76059	−0.880294	−	0.474428i	$-0.842655\pi$
$431$	−2.92636e9	−1.76059	−0.880294	+	0.474428i	$0.842655\pi$
$432$	−3.87441e8	−0.231213
$433$	−6.83606e8	−0.404668	−0.202334	−	0.979317i	$-0.564853\pi$
$433$	−6.83606e8	−0.404668	−0.202334	+	0.979317i	$0.564853\pi$
$434$	−6.13591e8	−0.360300
$435$	−4.82753e8	−0.281198
$436$	3.96246e8	0.228961
$437$	3.14433e9	1.80237
$438$	2.73052e8	0.155270
$439$	−9.63693e8	−0.543642	−0.271821	−	0.962348i	$-0.587626\pi$
$439$	−9.63693e8	−0.543642	−0.271821	+	0.962348i	$0.587626\pi$
$440$	8.93749e8	0.500186
$441$	−4.97456e8	−0.276197
$442$	1.47324e7	0.00811514
$443$	−1.22895e9	−0.671616	−0.335808	−	0.941930i	$-0.609009\pi$
$443$	−1.22895e9	−0.671616	−0.335808	+	0.941930i	$0.609009\pi$
$444$	−3.32178e8	−0.180107
$445$	−3.80624e9	−2.04756
$446$	1.22117e9	0.651786
$447$	1.20406e9	0.637632
$448$	−4.61674e8	−0.242584
$449$	2.87288e9	1.49781	0.748903	−	0.662679i	$-0.230581\pi$
$449$	2.87288e9	1.49781	0.748903	+	0.662679i	$0.230581\pi$
$450$	−8.03098e8	−0.415456
$451$	−1.07118e8	−0.0549848
$452$	3.59666e8	0.183196
$453$	1.04246e9	0.526887
$454$	−1.59462e8	−0.0799764
$455$	2.19141e7	0.0109065
$456$	−1.75074e9	−0.864657
$457$	5.34428e8	0.261929	0.130964	−	0.991387i	$-0.458193\pi$
$457$	5.34428e8	0.261929	0.130964	+	0.991387i	$0.458193\pi$
$458$	1.99112e9	0.968430
$459$	−1.57418e8	−0.0759817
$460$	8.29891e8	0.397529
$461$	−2.83200e9	−1.34629	−0.673146	−	0.739510i	$-0.735057\pi$
$461$	−2.83200e9	−1.34629	−0.673146	+	0.739510i	$0.735057\pi$
$462$	−2.42635e8	−0.114474
$463$	−2.92579e9	−1.36997	−0.684984	−	0.728558i	$-0.740191\pi$
$463$	−2.92579e9	−1.36997	−0.684984	+	0.728558i	$0.740191\pi$
$464$	−8.68525e8	−0.403617
$465$	1.39617e9	0.643950
$466$	2.03888e9	0.933343
$467$	−1.22471e9	−0.556449	−0.278225	−	0.960516i	$-0.589746\pi$
$467$	−1.22471e9	−0.556449	−0.278225	+	0.960516i	$0.589746\pi$
$468$	3.75513e6	0.00169342
$469$	−1.25291e8	−0.0560809
$470$	6.90258e7	0.0306668
$471$	1.82323e9	0.804023
$472$	2.42375e8	0.106094
$473$	−2.46015e8	−0.106893
$474$	−6.23970e8	−0.269116
$475$	−4.72963e9	−2.02488
$476$	1.07534e8	0.0457006
$477$	−4.68544e8	−0.197668
$478$	−3.65351e9	−1.53008
$479$	−1.72272e9	−0.716211	−0.358106	−	0.933681i	$-0.616577\pi$
$479$	−1.72272e9	−0.716211	−0.358106	+	0.933681i	$0.616577\pi$
$480$	−1.10348e9	−0.455429
$481$	−4.94827e7	−0.0202743
$482$	−2.66468e9	−1.08388
$483$	5.80530e8	0.234428
$484$	−5.72388e8	−0.229473
$485$	−3.13427e8	−0.124750
$486$	−1.83636e8	−0.0725657
$487$	2.39061e9	0.937901	0.468950	−	0.883225i	$-0.344632\pi$
$487$	2.39061e9	0.937901	0.468950	+	0.883225i	$0.344632\pi$
$488$	−1.46613e9	−0.571087
$489$	−1.00818e9	−0.389905
$490$	−3.53883e9	−1.35886
$491$	5.12479e8	0.195385	0.0976924	−	0.995217i	$-0.468854\pi$
$491$	5.12479e8	0.195385	0.0976924	+	0.995217i	$0.468854\pi$
$492$	5.53807e7	0.0209643
$493$	−3.52883e8	−0.132638
$494$	1.01213e8	0.0377740
$495$	5.52092e8	0.204594
$496$	2.51185e9	0.924291
$497$	−1.65641e9	−0.605231
$498$	−1.17255e9	−0.425431
$499$	3.61805e9	1.30354	0.651768	−	0.758418i	$-0.274028\pi$
$499$	3.61805e9	1.30354	0.651768	+	0.758418i	$0.274028\pi$
$500$	−1.15359e8	−0.0412720
$501$	1.10356e9	0.392070
$502$	4.35632e9	1.53694
$503$	3.85817e9	1.35174	0.675870	−	0.737021i	$-0.263768\pi$
$503$	3.85817e9	1.35174	0.675870	+	0.737021i	$0.263768\pi$
$504$	−3.23235e8	−0.112463
$505$	2.65140e8	0.0916125
$506$	−1.36878e9	−0.469685
$507$	−1.69365e9	−0.577160
$508$	−8.96722e8	−0.303483
$509$	5.17337e9	1.73885	0.869423	−	0.494068i	$-0.164491\pi$
$509$	5.17337e9	1.73885	0.869423	+	0.494068i	$0.164491\pi$
$510$	−1.11985e9	−0.373821
$511$	2.96894e8	0.0984300
$512$	9.88137e8	0.325366
$513$	−1.08148e9	−0.353676
$514$	−2.04533e9	−0.664345
$515$	−2.17122e9	−0.700453
$516$	1.27192e8	0.0407555
$517$	−2.48754e7	−0.00791686
$518$	−1.65303e9	−0.522547
$519$	1.06058e9	0.333010
$520$	−6.88331e7	−0.0214677
$521$	−5.01575e9	−1.55383	−0.776916	−	0.629604i	$-0.783217\pi$
$521$	−5.01575e9	−1.55383	−0.776916	+	0.629604i	$0.783217\pi$
$522$	−4.11657e8	−0.126674
$523$	3.72739e9	1.13933	0.569664	−	0.821878i	$-0.307073\pi$
$523$	3.72739e9	1.13933	0.569664	+	0.821878i	$0.307073\pi$
$524$	−4.62089e8	−0.140303
$525$	−8.73220e8	−0.263370
$526$	9.89832e8	0.296559
$527$	1.02057e9	0.303743
$528$	9.93273e8	0.293663
$529$	−1.29868e8	−0.0381422
$530$	−3.33316e9	−0.972501
$531$	1.49721e8	0.0433963
$532$	7.38769e8	0.212725
$533$	8.24977e6	0.00235992
$534$	−3.24569e9	−0.922386
$535$	2.00825e9	0.566995
$536$	3.93544e8	0.110387
$537$	−1.32090e9	−0.368095
$538$	5.72864e9	1.58604
$539$	1.27532e9	0.350798
$540$	−2.85436e8	−0.0780066
$541$	−7.61357e8	−0.206727	−0.103364	−	0.994644i	$-0.532961\pi$
$541$	−7.61357e8	−0.206727	−0.103364	+	0.994644i	$0.532961\pi$
$542$	4.34449e9	1.17204
$543$	−2.42426e9	−0.649799
$544$	−8.06622e8	−0.214820
$545$	−4.48676e9	−1.18726
$546$	1.86868e7	0.00491315
$547$	−1.94954e9	−0.509304	−0.254652	−	0.967033i	$-0.581961\pi$
$547$	−1.94954e9	−0.509304	−0.254652	+	0.967033i	$0.581961\pi$
$548$	1.15055e8	0.0298657
$549$	−9.05664e8	−0.233595
$550$	2.05888e9	0.527670
$551$	−2.42434e9	−0.617395
$552$	−1.82347e9	−0.461436
$553$	−6.78452e8	−0.170601
$554$	4.22811e9	1.05648
$555$	3.76130e9	0.933927
$556$	4.98602e8	0.123025
$557$	−4.11925e9	−1.01001	−0.505004	−	0.863117i	$-0.668509\pi$
$557$	−4.11925e9	−1.01001	−0.505004	+	0.863117i	$0.668509\pi$
$558$	1.19055e9	0.290087
$559$	1.89471e7	0.00458777
$560$	−2.99685e9	−0.721119
$561$	4.03568e8	0.0965044
$562$	7.16319e8	0.170227
$563$	−1.25233e9	−0.295760	−0.147880	−	0.989005i	$-0.547245\pi$
$563$	−1.25233e9	−0.295760	−0.147880	+	0.989005i	$0.547245\pi$
$564$	1.28608e7	0.00301850
$565$	−4.07256e9	−0.949943
$566$	4.38762e9	1.01712
$567$	−1.99670e8	−0.0460016
$568$	5.20286e9	1.19130
$569$	−1.82236e9	−0.414708	−0.207354	−	0.978266i	$-0.566485\pi$
$569$	−1.82236e9	−0.414708	−0.207354	+	0.978266i	$0.566485\pi$
$570$	−7.69346e9	−1.74004
$571$	−2.71123e9	−0.609453	−0.304727	−	0.952440i	$-0.598565\pi$
$571$	−2.71123e9	−0.609453	−0.304727	+	0.952440i	$0.598565\pi$
$572$	−9.62694e6	−0.00215081
$573$	3.17450e9	0.704910
$574$	2.75593e8	0.0608242
$575$	−4.92611e9	−1.08060
$576$	8.95787e8	0.195311
$577$	−6.72315e8	−0.145699	−0.0728496	−	0.997343i	$-0.523209\pi$
$577$	−6.72315e8	−0.145699	−0.0728496	+	0.997343i	$0.523209\pi$
$578$	4.43290e9	0.954862
$579$	−5.11862e9	−1.09592
$580$	−6.39862e8	−0.136172
$581$	−1.27493e9	−0.269693
$582$	−2.67268e8	−0.0561974
$583$	1.20120e9	0.251058
$584$	−9.32555e8	−0.193744
$585$	−4.25199e7	−0.00878106
$586$	−7.44603e9	−1.52856
$587$	−2.63538e9	−0.537786	−0.268893	−	0.963170i	$-0.586658\pi$
$587$	−2.63538e9	−0.537786	−0.268893	+	0.963170i	$0.586658\pi$
$588$	−6.59350e8	−0.133750
$589$	7.01142e9	1.41385
$590$	1.06510e9	0.213504
$591$	1.67202e9	0.333185
$592$	6.76699e9	1.34051
$593$	1.67193e9	0.329251	0.164625	−	0.986356i	$-0.447359\pi$
$593$	1.67193e9	0.329251	0.164625	+	0.986356i	$0.447359\pi$
$594$	4.70784e8	0.0921657
$595$	−1.21763e9	−0.236976
$596$	1.59591e9	0.308778
$597$	4.54883e9	0.874963
$598$	1.05418e8	0.0201586
$599$	1.58648e9	0.301607	0.150803	−	0.988564i	$-0.451814\pi$
$599$	1.58648e9	0.301607	0.150803	+	0.988564i	$0.451814\pi$
$600$	2.74282e9	0.518403
$601$	1.06586e9	0.200280	0.100140	−	0.994973i	$-0.468071\pi$
$601$	1.06586e9	0.200280	0.100140	+	0.994973i	$0.468071\pi$
$602$	6.32950e8	0.118245
$603$	2.43102e8	0.0451521
$604$	1.38173e9	0.255149
$605$	6.48124e9	1.18991
$606$	2.26092e8	0.0412697
$607$	9.08916e9	1.64954	0.824771	−	0.565467i	$-0.191304\pi$
$607$	9.08916e9	1.64954	0.824771	+	0.565467i	$0.191304\pi$
$608$	−5.54157e9	−0.999932
$609$	−4.47600e8	−0.0803027
$610$	−6.44276e9	−1.14926
$611$	1.91580e6	0.000339787 0
$612$	−2.08648e8	−0.0367947
$613$	−3.73946e9	−0.655688	−0.327844	−	0.944732i	$-0.606322\pi$
$613$	−3.73946e9	−0.655688	−0.327844	+	0.944732i	$0.606322\pi$
$614$	7.95797e9	1.38744
$615$	−6.27085e8	−0.108709
$616$	8.28669e8	0.142840
$617$	5.49157e9	0.941236	0.470618	−	0.882337i	$-0.344031\pi$
$617$	5.49157e9	0.941236	0.470618	+	0.882337i	$0.344031\pi$
$618$	−1.85146e9	−0.315540
$619$	6.71900e9	1.13864	0.569321	−	0.822115i	$-0.307206\pi$
$619$	6.71900e9	1.13864	0.569321	+	0.822115i	$0.307206\pi$
$620$	1.85054e9	0.311837
$621$	−1.12640e9	−0.188744
$622$	1.00593e10	1.67610
$623$	−3.52908e9	−0.584728
$624$	−7.64980e7	−0.0126039
$625$	−5.41876e9	−0.887810
$626$	4.76915e9	0.777018
$627$	2.77255e9	0.449204
$628$	2.41659e9	0.389354
$629$	2.74944e9	0.440521
$630$	−1.42043e9	−0.226322
$631$	3.57282e9	0.566119	0.283060	−	0.959102i	$-0.408651\pi$
$631$	3.57282e9	0.566119	0.283060	+	0.959102i	$0.408651\pi$
$632$	2.13104e9	0.335801
$633$	5.13155e8	0.0804148
$634$	−3.04765e9	−0.474955
$635$	1.01537e10	1.57368
$636$	−6.21029e8	−0.0957221
$637$	−9.82199e7	−0.0150561
$638$	1.05536e9	0.160889
$639$	3.21394e9	0.487287
$640$	1.16038e10	1.74973
$641$	−9.37478e8	−0.140591	−0.0702956	−	0.997526i	$-0.522394\pi$
$641$	−9.37478e8	−0.140591	−0.0702956	+	0.997526i	$0.522394\pi$
$642$	1.71249e9	0.255420
$643$	8.40132e9	1.24626	0.623131	−	0.782118i	$-0.285860\pi$
$643$	8.40132e9	1.24626	0.623131	+	0.782118i	$0.285860\pi$
$644$	7.69461e8	0.113524
$645$	−1.44022e9	−0.211334
$646$	−5.62377e9	−0.820755
$647$	1.33915e10	1.94386	0.971931	−	0.235264i	$-0.0755956\pi$
$647$	1.33915e10	1.94386	0.971931	+	0.235264i	$0.0755956\pi$
$648$	6.27172e8	0.0905471
$649$	−3.83837e8	−0.0551176
$650$	−1.58567e8	−0.0226473
$651$	1.29450e9	0.183895
$652$	−1.33629e9	−0.188814
$653$	−7.85725e9	−1.10427	−0.552134	−	0.833756i	$-0.686186\pi$
$653$	−7.85725e9	−1.10427	−0.552134	+	0.833756i	$0.686186\pi$
$654$	−3.82599e9	−0.534837
$655$	5.23231e9	0.727526
$656$	−1.12819e9	−0.156034
$657$	−5.76063e8	−0.0792485
$658$	6.39995e7	0.00875762
$659$	3.05094e9	0.415273	0.207637	−	0.978206i	$-0.433423\pi$
$659$	3.05094e9	0.415273	0.207637	+	0.978206i	$0.433423\pi$
$660$	7.31767e8	0.0990761
$661$	1.41136e10	1.90079	0.950395	−	0.311045i	$-0.100679\pi$
$661$	1.41136e10	1.90079	0.950395	+	0.311045i	$0.100679\pi$
$662$	−8.31838e9	−1.11439
$663$	−3.10812e7	−0.00414191
$664$	4.00461e9	0.530850
$665$	−8.36521e9	−1.10306
$666$	3.20737e9	0.420716
$667$	−2.52506e9	−0.329481
$668$	1.46270e9	0.189862
$669$	−2.57633e9	−0.332667
$670$	1.72939e9	0.222143
$671$	2.32183e9	0.296689
$672$	−1.02313e9	−0.130058
$673$	1.56465e10	1.97863	0.989313	−	0.145809i	$-0.0465786\pi$
$673$	1.56465e10	1.97863	0.989313	+	0.145809i	$0.0465786\pi$
$674$	−8.51144e9	−1.07076
$675$	1.69431e9	0.212046
$676$	−2.24484e9	−0.279494
$677$	5.00729e9	0.620215	0.310107	−	0.950702i	$-0.399635\pi$
$677$	5.00729e9	0.620215	0.310107	+	0.950702i	$0.399635\pi$
$678$	−3.47278e9	−0.427931
$679$	−2.90604e8	−0.0356252
$680$	3.82461e9	0.466451
$681$	3.36420e8	0.0408194
$682$	−3.05219e9	−0.368439
$683$	−1.48727e10	−1.78615	−0.893073	−	0.449911i	$-0.851456\pi$
$683$	−1.48727e10	−1.78615	−0.893073	+	0.449911i	$0.851456\pi$
$684$	−1.43343e9	−0.171270
$685$	−1.30278e9	−0.154866
$686$	−7.24105e9	−0.856381
$687$	−4.20070e9	−0.494280
$688$	−2.59110e9	−0.303337
$689$	−9.25114e7	−0.0107753
$690$	−8.01308e9	−0.928597
$691$	−1.34895e8	−0.0155532	−0.00777662	−	0.999970i	$-0.502475\pi$
$691$	−1.34895e8	−0.0155532	−0.00777662	+	0.999970i	$0.502475\pi$
$692$	1.40574e9	0.161263
$693$	5.11890e8	0.0584266
$694$	−3.40507e9	−0.386695
$695$	−5.64576e9	−0.637933
$696$	1.40593e9	0.158064
$697$	−4.58387e8	−0.0512764
$698$	−1.54593e10	−1.72066
$699$	−4.30146e9	−0.476372
$700$	−1.15740e9	−0.127539
$701$	−8.34830e9	−0.915345	−0.457672	−	0.889121i	$-0.651317\pi$
$701$	−8.34830e9	−0.915345	−0.457672	+	0.889121i	$0.651317\pi$
$702$	−3.62579e7	−0.00395570
$703$	1.88889e10	2.05052
$704$	−2.29651e9	−0.248064
$705$	−1.45625e8	−0.0156521
$706$	−3.04982e9	−0.326180
$707$	2.45833e8	0.0261621
$708$	1.98447e8	0.0210150
$709$	−1.42535e9	−0.150197	−0.0750983	−	0.997176i	$-0.523927\pi$
$709$	−1.42535e9	−0.150197	−0.0750983	+	0.997176i	$0.523927\pi$
$710$	2.28635e10	2.39739
$711$	1.31640e9	0.137355
$712$	1.10850e10	1.15095
$713$	7.30270e9	0.754519
$714$	−1.03830e9	−0.106753
$715$	1.09007e8	0.0111528
$716$	−1.75078e9	−0.178253
$717$	7.70788e9	0.780940
$718$	1.31848e10	1.32935
$719$	−5.22232e9	−0.523978	−0.261989	−	0.965071i	$-0.584378\pi$
$719$	−5.22232e9	−0.523978	−0.261989	+	0.965071i	$0.584378\pi$
$720$	5.81480e9	0.580592
$721$	−2.01312e9	−0.200031
$722$	−2.71961e10	−2.68922
$723$	5.62172e9	0.553204
$724$	−3.21322e9	−0.314670
$725$	3.79813e9	0.370158
$726$	5.52674e9	0.536032
$727$	1.04981e10	1.01330	0.506652	−	0.862150i	$-0.330883\pi$
$727$	1.04981e10	1.01330	0.506652	+	0.862150i	$0.330883\pi$
$728$	−6.38208e7	−0.00613060
$729$	3.87420e8	0.0370370
$730$	−4.09803e9	−0.389893
$731$	−1.05277e9	−0.0996833
$732$	−1.20041e9	−0.113120
$733$	−1.91930e10	−1.80003	−0.900014	−	0.435862i	$-0.856444\pi$
$733$	−1.91930e10	−1.80003	−0.900014	+	0.435862i	$0.856444\pi$
$734$	−1.92426e10	−1.79608
$735$	7.46593e9	0.693551
$736$	−5.77179e9	−0.533628
$737$	−6.23235e8	−0.0573477
$738$	−5.34733e8	−0.0489711
$739$	−1.50672e10	−1.37334	−0.686669	−	0.726970i	$-0.740928\pi$
$739$	−1.50672e10	−1.37334	−0.686669	+	0.726970i	$0.740928\pi$
$740$	4.98539e9	0.452260
$741$	−2.13531e8	−0.0192796
$742$	−3.09045e9	−0.277720
$743$	−1.35355e10	−1.21064	−0.605318	−	0.795984i	$-0.706954\pi$
$743$	−1.35355e10	−1.21064	−0.605318	+	0.795984i	$0.706954\pi$
$744$	−4.06608e9	−0.361969
$745$	−1.80707e10	−1.60114
$746$	7.07975e9	0.624355
$747$	2.47375e9	0.217137
$748$	5.34907e8	0.0467329
$749$	1.86201e9	0.161918
$750$	1.11385e9	0.0964082
$751$	−4.44209e9	−0.382690	−0.191345	−	0.981523i	$-0.561285\pi$
$751$	−4.44209e9	−0.382690	−0.191345	+	0.981523i	$0.561285\pi$
$752$	−2.61995e8	−0.0224662
$753$	−9.19061e9	−0.784444
$754$	−8.12793e7	−0.00690527
$755$	−1.56455e10	−1.32305
$756$	−2.64652e8	−0.0222766
$757$	−1.96167e10	−1.64358	−0.821789	−	0.569792i	$-0.807024\pi$
$757$	−1.96167e10	−1.64358	−0.821789	+	0.569792i	$0.807024\pi$
$758$	−1.35186e10	−1.12743
$759$	2.88774e9	0.239724
$760$	2.62755e10	2.17121
$761$	−8.13676e9	−0.669276	−0.334638	−	0.942347i	$-0.608614\pi$
$761$	−8.13676e9	−0.669276	−0.334638	+	0.942347i	$0.608614\pi$
$762$	8.65837e9	0.708914
$763$	−4.16005e9	−0.339049
$764$	4.20762e9	0.341357
$765$	2.36256e9	0.190795
$766$	1.33506e9	0.107325
$767$	2.95616e7	0.00236562
$768$	5.64820e9	0.449931
$769$	−1.79213e10	−1.42111	−0.710554	−	0.703642i	$-0.751556\pi$
$769$	−1.79213e10	−1.42111	−0.710554	+	0.703642i	$0.751556\pi$
$770$	3.64151e9	0.287451
$771$	4.31508e9	0.339077
$772$	−6.78444e9	−0.530705
$773$	1.39926e10	1.08961	0.544804	−	0.838563i	$-0.316604\pi$
$773$	1.39926e10	1.08961	0.544804	+	0.838563i	$0.316604\pi$
$774$	−1.22811e9	−0.0952017
$775$	−1.09845e10	−0.847669
$776$	9.12799e8	0.0701228
$777$	3.48741e9	0.266704
$778$	1.95023e10	1.48476
$779$	−3.14916e9	−0.238679
$780$	−5.63578e7	−0.00425229
$781$	−8.23950e9	−0.618903
$782$	−5.85740e9	−0.438007
$783$	8.68480e8	0.0646537
$784$	1.34320e10	0.995485
$785$	−2.73635e10	−2.01896
$786$	4.46174e9	0.327737
$787$	7.87847e9	0.576143	0.288071	−	0.957609i	$-0.406986\pi$
$787$	7.87847e9	0.576143	0.288071	+	0.957609i	$0.406986\pi$
$788$	2.21617e9	0.161347
$789$	−2.08827e9	−0.151362
$790$	9.36468e9	0.675769
$791$	−3.77601e9	−0.271278
$792$	−1.60787e9	−0.115004
$793$	−1.78818e8	−0.0127337
$794$	3.55298e9	0.251896
$795$	7.03202e9	0.496358
$796$	6.02922e9	0.423707
$797$	−7.12267e9	−0.498355	−0.249178	−	0.968458i	$-0.580160\pi$
$797$	−7.12267e9	−0.498355	−0.249178	+	0.968458i	$0.580160\pi$
$798$	−7.13325e9	−0.496909
$799$	−1.06449e8	−0.00738291
$800$	8.68179e9	0.599507
$801$	6.84749e9	0.470779
$802$	−4.34517e9	−0.297438
$803$	1.47684e9	0.100653
$804$	3.22218e8	0.0218652
$805$	−8.71273e9	−0.588666
$806$	2.35067e8	0.0158132
$807$	−1.20858e10	−0.809503
$808$	−7.72171e8	−0.0514960
$809$	−1.13613e10	−0.754412	−0.377206	−	0.926129i	$-0.623115\pi$
$809$	−1.13613e10	−0.754412	−0.377206	+	0.926129i	$0.623115\pi$
$810$	2.75605e9	0.182218
$811$	−1.58411e10	−1.04283	−0.521413	−	0.853304i	$-0.674595\pi$
$811$	−1.58411e10	−1.04283	−0.521413	+	0.853304i	$0.674595\pi$
$812$	−5.93269e8	−0.0388871
$813$	−9.16565e9	−0.598200
$814$	−8.22265e9	−0.534351
$815$	1.51311e10	0.979079
$816$	4.25050e9	0.273857
$817$	−7.23263e9	−0.464001
$818$	−2.86332e10	−1.82908
$819$	−3.94238e7	−0.00250764
$820$	−8.31166e8	−0.0526429
$821$	2.50210e10	1.57799	0.788993	−	0.614402i	$-0.210603\pi$
$821$	2.50210e10	1.57799	0.788993	+	0.614402i	$0.210603\pi$
$822$	−1.11092e9	−0.0697640
$823$	1.51409e10	0.946790	0.473395	−	0.880850i	$-0.343028\pi$
$823$	1.51409e10	0.946790	0.473395	+	0.880850i	$0.343028\pi$
$824$	6.32329e9	0.393730
$825$	−4.34366e9	−0.269319
$826$	9.87539e8	0.0609711
$827$	−7.61883e9	−0.468402	−0.234201	−	0.972188i	$-0.575247\pi$
$827$	−7.61883e9	−0.468402	−0.234201	+	0.972188i	$0.575247\pi$
$828$	−1.49299e9	−0.0914007
$829$	−1.04307e10	−0.635879	−0.317940	−	0.948111i	$-0.602991\pi$
$829$	−1.04307e10	−0.635879	−0.317940	+	0.948111i	$0.602991\pi$
$830$	1.75979e10	1.06829
$831$	−8.92011e9	−0.539221
$832$	1.76868e8	0.0106468
$833$	5.45745e9	0.327139
$834$	−4.81429e9	−0.287377
$835$	−1.65624e10	−0.984514
$836$	3.67486e9	0.217530
$837$	−2.51172e9	−0.148058
$838$	−2.20617e10	−1.29505
$839$	−8.81723e9	−0.515425	−0.257712	−	0.966222i	$-0.582969\pi$
$839$	−8.81723e9	−0.515425	−0.257712	+	0.966222i	$0.582969\pi$
$840$	4.85118e9	0.282403
$841$	−1.53030e10	−0.887137
$842$	−3.03804e10	−1.75388
$843$	−1.51123e9	−0.0868829
$844$	6.80159e8	0.0389414
$845$	2.54187e10	1.44929
$846$	−1.24178e8	−0.00705098
$847$	6.00930e9	0.339807
$848$	1.26514e10	0.712445
$849$	−9.25664e9	−0.519130
$850$	8.81056e9	0.492082
$851$	1.96736e10	1.09429
$852$	4.25990e9	0.235972
$853$	6.88811e9	0.379995	0.189998	−	0.981785i	$-0.439152\pi$
$853$	6.88811e9	0.379995	0.189998	+	0.981785i	$0.439152\pi$
$854$	−5.97362e9	−0.328197
$855$	1.62310e10	0.888105
$856$	−5.84866e9	−0.318712
$857$	7.74166e9	0.420147	0.210074	−	0.977686i	$-0.432630\pi$
$857$	7.74166e9	0.420147	0.210074	+	0.977686i	$0.432630\pi$
$858$	9.29537e7	0.00502413
$859$	−7.99776e9	−0.430519	−0.215259	−	0.976557i	$-0.569060\pi$
$859$	−7.99776e9	−0.430519	−0.215259	+	0.976557i	$0.569060\pi$
$860$	−1.90893e9	−0.102340
$861$	−5.81423e8	−0.0310442
$862$	3.74514e10	1.99156
$863$	−2.16260e10	−1.14535	−0.572675	−	0.819782i	$-0.694094\pi$
$863$	−2.16260e10	−1.14535	−0.572675	+	0.819782i	$0.694094\pi$
$864$	1.98518e9	0.104713
$865$	−1.59174e10	−0.836212
$866$	8.74875e9	0.457755
$867$	−9.35216e9	−0.487355
$868$	1.71579e9	0.0890523
$869$	−3.37483e9	−0.174454
$870$	6.17824e9	0.318088
$871$	4.79991e7	0.00246133
$872$	1.30669e10	0.667366
$873$	5.63859e8	0.0286828
$874$	−4.02409e10	−2.03882
$875$	1.21111e9	0.0611160
$876$	−7.63539e8	−0.0383766
$877$	−2.67027e10	−1.33677	−0.668385	−	0.743816i	$-0.733014\pi$
$877$	−2.67027e10	−1.33677	−0.668385	+	0.743816i	$0.733014\pi$
$878$	1.23333e10	0.614961
$879$	1.57090e10	0.780167
$880$	−1.49073e10	−0.737409
$881$	−1.15396e10	−0.568559	−0.284279	−	0.958741i	$-0.591754\pi$
$881$	−1.15396e10	−0.568559	−0.284279	+	0.958741i	$0.591754\pi$
$882$	6.36641e9	0.312431
$883$	−3.23054e10	−1.57911	−0.789555	−	0.613679i	$-0.789689\pi$
$883$	−3.23054e10	−1.57911	−0.789555	+	0.613679i	$0.789689\pi$
$884$	−4.11964e7	−0.00200575
$885$	−2.24705e9	−0.108971
$886$	1.57280e10	0.759724
$887$	1.57129e10	0.756005	0.378002	−	0.925805i	$-0.376611\pi$
$887$	1.57129e10	0.756005	0.378002	+	0.925805i	$0.376611\pi$
$888$	−1.09541e10	−0.524967
$889$	9.41436e9	0.449402
$890$	4.87120e10	2.31617
$891$	−9.93221e8	−0.0470407
$892$	−3.41478e9	−0.161096
$893$	−7.31314e8	−0.0343656
$894$	−1.54094e10	−0.721282
$895$	1.98244e10	0.924312
$896$	1.07589e10	0.499676
$897$	−2.22402e8	−0.0102888
$898$	−3.67670e10	−1.69430
$899$	−5.63053e9	−0.258458
$900$	2.24571e9	0.102685
$901$	5.14026e9	0.234125
$902$	1.37088e9	0.0621982
$903$	−1.33534e9	−0.0603512
$904$	1.18606e10	0.533970
$905$	3.63838e10	1.63169
$906$	−1.33414e10	−0.596008
$907$	−6.98597e9	−0.310886	−0.155443	−	0.987845i	$-0.549681\pi$
$907$	−6.98597e9	−0.310886	−0.155443	+	0.987845i	$0.549681\pi$
$908$	4.45905e8	0.0197671
$909$	−4.76990e8	−0.0210637
$910$	−2.80455e8	−0.0123372
$911$	−4.32364e9	−0.189468	−0.0947339	−	0.995503i	$-0.530200\pi$
$911$	−4.32364e9	−0.189468	−0.0947339	+	0.995503i	$0.530200\pi$
$912$	2.92014e10	1.27474
$913$	−6.34189e9	−0.275785
$914$	−6.83958e9	−0.296290
$915$	1.35924e10	0.586573
$916$	−5.56779e9	−0.239358
$917$	4.85131e9	0.207762
$918$	2.01462e9	0.0859496
$919$	−2.41430e10	−1.02609	−0.513046	−	0.858361i	$-0.671483\pi$
$919$	−2.41430e10	−1.02609	−0.513046	+	0.858361i	$0.671483\pi$
$920$	2.73670e10	1.15870
$921$	−1.67891e10	−0.708138
$922$	3.62437e10	1.52291
$923$	6.34574e8	0.0265630
$924$	6.78482e8	0.0282935
$925$	−2.95926e10	−1.22938
$926$	3.74441e10	1.54969
$927$	3.90606e9	0.161050
$928$	4.45017e9	0.182792
$929$	2.15464e10	0.881699	0.440850	−	0.897581i	$-0.354677\pi$
$929$	2.15464e10	0.881699	0.440850	+	0.897581i	$0.354677\pi$
$930$	−1.78680e10	−0.728428
$931$	3.74932e10	1.52275
$932$	−5.70135e9	−0.230686
$933$	−2.12222e10	−0.855471
$934$	1.56738e10	0.629449
$935$	−6.05684e9	−0.242329
$936$	1.23832e8	0.00493590
$937$	3.83047e10	1.52112	0.760561	−	0.649267i	$-0.224924\pi$
$937$	3.83047e10	1.52112	0.760561	+	0.649267i	$0.224924\pi$
$938$	1.60346e9	0.0634380
$939$	−1.00616e10	−0.396584
$940$	−1.93018e8	−0.00757966
$941$	−2.56605e10	−1.00392	−0.501962	−	0.864890i	$-0.667388\pi$
$941$	−2.56605e10	−1.00392	−0.501962	+	0.864890i	$0.667388\pi$
$942$	−2.33336e10	−0.909502
$943$	−3.27999e9	−0.127374
$944$	−4.04269e9	−0.156411
$945$	2.99670e9	0.115513
$946$	3.14848e9	0.120916
$947$	4.39924e10	1.68326	0.841632	−	0.540051i	$-0.181595\pi$
$947$	4.39924e10	1.68326	0.841632	+	0.540051i	$0.181595\pi$
$948$	1.74481e9	0.0665151
$949$	−1.13740e8	−0.00431999
$950$	6.05294e10	2.29052
$951$	6.42967e9	0.242414
$952$	3.54611e9	0.133206
$953$	−4.24635e8	−0.0158924	−0.00794622	−	0.999968i	$-0.502529\pi$
$953$	−4.24635e8	−0.0158924	−0.00794622	+	0.999968i	$0.502529\pi$
$954$	5.99640e9	0.223600
$955$	−4.76435e10	−1.77008
$956$	1.02164e10	0.378176
$957$	−2.22650e9	−0.0821166
$958$	2.20473e10	0.810170
$959$	−1.20792e9	−0.0442255
$960$	−1.34442e10	−0.490439
$961$	−1.12286e10	−0.408126
$962$	6.33276e8	0.0229340
$963$	−3.61287e9	−0.130365
$964$	7.45127e9	0.267893
$965$	7.68213e10	2.75192
$966$	−7.42959e9	−0.265182
$967$	4.76260e10	1.69376	0.846879	−	0.531786i	$-0.178479\pi$
$967$	4.76260e10	1.69376	0.846879	+	0.531786i	$0.178479\pi$
$968$	−1.88755e10	−0.668857
$969$	1.18646e10	0.418908
$970$	4.01122e9	0.141116
$971$	−1.01771e10	−0.356744	−0.178372	−	0.983963i	$-0.557083\pi$
$971$	−1.01771e10	−0.356744	−0.178372	+	0.983963i	$0.557083\pi$
$972$	5.13504e8	0.0179354
$973$	−5.23465e9	−0.182177
$974$	−3.05948e10	−1.06094
$975$	3.34532e8	0.0115590
$976$	2.44542e10	0.841936
$977$	5.07713e10	1.74176	0.870878	−	0.491499i	$-0.163551\pi$
$977$	5.07713e10	1.74176	0.870878	+	0.491499i	$0.163551\pi$
$978$	1.29027e10	0.441056
$979$	−1.75547e10	−0.597937
$980$	9.89567e9	0.335857
$981$	8.07174e9	0.272977
$982$	−6.55867e9	−0.221017
$983$	3.69088e10	1.23934	0.619672	−	0.784861i	$-0.287266\pi$
$983$	3.69088e10	1.23934	0.619672	+	0.784861i	$0.287266\pi$
$984$	1.82627e9	0.0611058
$985$	−2.50941e10	−0.836651
$986$	4.51617e9	0.150038
$987$	−1.35021e8	−0.00446983
$988$	−2.83024e8	−0.00933627
$989$	−7.53310e9	−0.247621
$990$	−7.06563e9	−0.231434
$991$	5.58119e10	1.82167	0.910833	−	0.412774i	$-0.135440\pi$
$991$	5.58119e10	1.82167	0.910833	+	0.412774i	$0.135440\pi$
$992$	−1.28703e10	−0.418598
$993$	1.75494e10	0.568775
$994$	2.11987e10	0.684630
$995$	−6.82699e10	−2.19709
$996$	3.27882e9	0.105150
$997$	−4.81978e9	−0.154026	−0.0770130	−	0.997030i	$-0.524538\pi$
$997$	−4.81978e9	−0.154026	−0.0770130	+	0.997030i	$0.524538\pi$
$998$	−4.63036e10	−1.47454
$999$	−6.76663e9	−0.214730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.d.1.5	✓	18
3.2	odd	2		531.8.a.e.1.14		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.5	✓	18	1.1	even	1	trivial
531.8.a.e.1.14		18	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-12.7979 q^{2} +27.0000 q^{3} +35.7870 q^{4} -405.222 q^{5} -345.544 q^{6} -375.715 q^{7} +1180.14 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-12.7979 q^{2} +27.0000 q^{3} +35.7870 q^{4} -405.222 q^{5} -345.544 q^{6} -375.715 q^{7} +1180.14 q^{8} +729.000 q^{9} +5186.00 q^{10} -1868.92 q^{11} +966.249 q^{12} +143.937 q^{13} +4808.37 q^{14} -10941.0 q^{15} -19684.0 q^{16} -7997.65 q^{17} -9329.69 q^{18} -54944.6 q^{19} -14501.7 q^{20} -10144.3 q^{21} +23918.3 q^{22} -57227.2 q^{23} +31863.7 q^{24} +86079.9 q^{25} -1842.09 q^{26} +19683.0 q^{27} -13445.7 q^{28} +44123.3 q^{29} +140022. q^{30} -127609. q^{31} +100857. q^{32} -50460.9 q^{33} +102353. q^{34} +152248. q^{35} +26088.7 q^{36} -343781. q^{37} +703178. q^{38} +3886.30 q^{39} -478217. q^{40} +57315.2 q^{41} +129826. q^{42} +131635. q^{43} -66883.0 q^{44} -295407. q^{45} +732390. q^{46} +13310.0 q^{47} -531469. q^{48} -682381. q^{49} -1.10164e6 q^{50} -215937. q^{51} +5151.07 q^{52} -642722. q^{53} -251902. q^{54} +757328. q^{55} -443395. q^{56} -1.48351e6 q^{57} -564687. q^{58} +205379. q^{59} -391545. q^{60} -1.24234e6 q^{61} +1.63313e6 q^{62} -273896. q^{63} +1.22879e6 q^{64} -58326.4 q^{65} +645794. q^{66} +333473. q^{67} -286212. q^{68} -1.54514e6 q^{69} -1.94846e6 q^{70} +4.40870e6 q^{71} +860319. q^{72} -790210. q^{73} +4.39968e6 q^{74} +2.32416e6 q^{75} -1.96630e6 q^{76} +702181. q^{77} -49736.5 q^{78} +1.80576e6 q^{79} +7.97640e6 q^{80} +531441. q^{81} -733516. q^{82} +3.39335e6 q^{83} -363034. q^{84} +3.24082e6 q^{85} -1.68465e6 q^{86} +1.19133e6 q^{87} -2.20558e6 q^{88} +9.39299e6 q^{89} +3.78060e6 q^{90} -54079.2 q^{91} -2.04799e6 q^{92} -3.44544e6 q^{93} -170341. q^{94} +2.22648e7 q^{95} +2.72315e6 q^{96} +773470. q^{97} +8.73307e6 q^{98} -1.36244e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9}+O(q^{10}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9} + 3609 q^{10} + 15070 q^{11} + 36666 q^{12} + 13662 q^{13} + 20861 q^{14} + 18306 q^{15} + 60482 q^{16} + 71919 q^{17} + 17496 q^{18} + 56231 q^{19} + 143053 q^{20} + 83187 q^{21} + 274198 q^{22} + 150029 q^{23} + 110889 q^{24} + 399672 q^{25} + 182846 q^{26} + 354294 q^{27} + 434150 q^{28} + 591285 q^{29} + 97443 q^{30} + 426733 q^{31} + 1205630 q^{32} + 406890 q^{33} + 403548 q^{34} + 912879 q^{35} + 989982 q^{36} + 7703 q^{37} - 417859 q^{38} + 368874 q^{39} + 618020 q^{40} + 770959 q^{41} + 563247 q^{42} + 793050 q^{43} + 2591274 q^{44} + 494262 q^{45} - 4068019 q^{46} + 1410373 q^{47} + 1633014 q^{48} + 1637427 q^{49} + 1021549 q^{50} + 1941813 q^{51} - 3749190 q^{52} + 1037934 q^{53} + 472392 q^{54} + 331974 q^{55} - 391748 q^{56} + 1518237 q^{57} + 653724 q^{58} + 3696822 q^{59} + 3862431 q^{60} - 1374623 q^{61} + 5251718 q^{62} + 2246049 q^{63} + 5077197 q^{64} + 3257170 q^{65} + 7403346 q^{66} - 2436904 q^{67} + 14119909 q^{68} + 4050783 q^{69} + 5185580 q^{70} + 14289172 q^{71} + 2994003 q^{72} + 5482515 q^{73} + 14934154 q^{74} + 10791144 q^{75} + 3822912 q^{76} + 23157109 q^{77} + 4936842 q^{78} + 19786414 q^{79} + 31978143 q^{80} + 9565938 q^{81} + 9749509 q^{82} + 30227337 q^{83} + 11722050 q^{84} + 9946981 q^{85} + 44295864 q^{86} + 15964695 q^{87} + 39970897 q^{88} + 31061677 q^{89} + 2630961 q^{90} + 26377785 q^{91} + 4719698 q^{92} + 11521791 q^{93} + 44488296 q^{94} + 15534599 q^{95} + 32552010 q^{96} + 12084118 q^{97} + 42274744 q^{98} + 10986030 q^{99}+O(q^{100}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9 + 3609 * q^10 + 15070 * q^11 + 36666 * q^12 + 13662 * q^13 + 20861 * q^14 + 18306 * q^15 + 60482 * q^16 + 71919 * q^17 + 17496 * q^18 + 56231 * q^19 + 143053 * q^20 + 83187 * q^21 + 274198 * q^22 + 150029 * q^23 + 110889 * q^24 + 399672 * q^25 + 182846 * q^26 + 354294 * q^27 + 434150 * q^28 + 591285 * q^29 + 97443 * q^30 + 426733 * q^31 + 1205630 * q^32 + 406890 * q^33 + 403548 * q^34 + 912879 * q^35 + 989982 * q^36 + 7703 * q^37 - 417859 * q^38 + 368874 * q^39 + 618020 * q^40 + 770959 * q^41 + 563247 * q^42 + 793050 * q^43 + 2591274 * q^44 + 494262 * q^45 - 4068019 * q^46 + 1410373 * q^47 + 1633014 * q^48 + 1637427 * q^49 + 1021549 * q^50 + 1941813 * q^51 - 3749190 * q^52 + 1037934 * q^53 + 472392 * q^54 + 331974 * q^55 - 391748 * q^56 + 1518237 * q^57 + 653724 * q^58 + 3696822 * q^59 + 3862431 * q^60 - 1374623 * q^61 + 5251718 * q^62 + 2246049 * q^63 + 5077197 * q^64 + 3257170 * q^65 + 7403346 * q^66 - 2436904 * q^67 + 14119909 * q^68 + 4050783 * q^69 + 5185580 * q^70 + 14289172 * q^71 + 2994003 * q^72 + 5482515 * q^73 + 14934154 * q^74 + 10791144 * q^75 + 3822912 * q^76 + 23157109 * q^77 + 4936842 * q^78 + 19786414 * q^79 + 31978143 * q^80 + 9565938 * q^81 + 9749509 * q^82 + 30227337 * q^83 + 11722050 * q^84 + 9946981 * q^85 + 44295864 * q^86 + 15964695 * q^87 + 39970897 * q^88 + 31061677 * q^89 + 2630961 * q^90 + 26377785 * q^91 + 4719698 * q^92 + 11521791 * q^93 + 44488296 * q^94 + 15534599 * q^95 + 32552010 * q^96 + 12084118 * q^97 + 42274744 * q^98 + 10986030 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more